Question

The chord of the circle x² + y² = a²  touches the rectangular hyperbola x² - y² = a². . Show that the locus of the midpoint of the chord is (x² + y²)² = a²(x² - y²).

Answer 1

The equation of the chord of  x² + y²  = a²  with (x₁, y₁) = x₁² + y²₁ as its midpoint is xx₁ + yy₁ =  x₁² +  y₁².  This chord touches the hyperbola x² - y² = a². So

 

Hence, the locus of (x₁,y₁) is (x² + y²)² = a²(x² - y²).